After staring at the videos repeatedly I now see that the Fourier series can be obtained mechanically for all the (tiles of the) curves. We have a complex basis B for the corresponding numeration system for our curve of order R, where abs(B) = sqrt(R) The lowest nontrivial approximation (let's call it "principal") comes from moving along the first iterate of the tile (Theta_{1} in my draft). The higher approximations are obtained by adding powers of (1/B) times the (possibly phase-shifted) principal played at R times the speed of the principal. Now that should be a geometric series, where telescoping is less of a miracle. Did I miss something? Best regards, jj P.S.: Looks like the arrangement of rods with rotating joints in the videos are the contributions of the principal (innermost rod) followed by the higher harmonics. Right? * rwg <rwg@sdf.org> [Dec 28. 2015 12:09]:
On 2015-12-27 22:36, Joerg Arndt wrote: [...]
Thank you for persisting!
Ahah, http://gosper.org/TDrag3c.mp4 is a Fudgeflake filled by three Triadic Dragons, m=3 in http://gosper.org/fst.pdf , p144 (actual page 16) et seq. The Fourier coefficients are infinite products of ordinary scalars. A technique similar to Julian's amazing little piecewiserecursivefractal function permits exact evaluation at any rational time value. Plugging said time into the Fourier series and equating produces closed forms for sums of infinite products which I think would have even gotten a rise out of Ramanujan. Earlier in the paper this is exhibited for the Snowflake family of fractals, leading to the pair of identities (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , where the only difference on the lhs is changing 260 to 261.
It's a shame that our unfamiliarity with matrix products makes it hard to appreciate the analogous identities for the dyadic (Heighway) Dragon and Sierpinski Gasket. As explained in fst.pdf, the Snowflake and Terdragon Fourier series were also derived with matrix products, which then miraculously telescoped. --rwg
[...]